1 Introduction and main results↩︎

We start this manuscript by defining a class of functions that are closely related to Dirichlet characters. These functions, that are going to be recurrent in what follows, are called modified Dirichlet characters, since they agree with some character \(\chi\) in all, but a finite subset of prime numbers. Precisely, we have the following:

We are concerned with the special case where we take \(\chi\) to be a real non-principal Dirichlet character of modulus \(q\). As in previous papers on the topic, we choose to denote our modified character by \(g_\chi\) and define it as \[\begin{align} g_\chi(p)= \begin{cases} \chi(p) &\text{ if p\nmid q,}\\ 1 &\text{ if p\mid q.} \end{cases} \end{align}\]

That is, our modification set \(\mathcal{S}\) consists of primes \(p\) that divides the modulus \(q\).

Moreover, let \(\mu^{(k)}\) be the indicator function of the \(k\)-free integers.

Recently, [1] proposed the question of finding an example of a multiplicative function \(f:\mathbb{N}\to\{-1,0,+1\}\) that resembles² the Möbius function and such that its partial sums have cancellation greater than square-root. In his investigation, the author proved that \[\sum_{n\leq x}\mu^{(2)}(n)g_\chi(n)\ll x^{\frac{2}{5}+\varepsilon},\forall\varepsilon>0,\] assuming the Riemann hypothesis.

Later, this result was generalized to the \(k\)-free integers by Aymone, Medeiros and the author [2]. Under the Riemann hypothesis for \(L\)-functions (or generalized Riemann hypothesis) we have \[\sum_{n\leq x}\mu^{(k)}(n)g_\chi(n)\ll x^{1/(k+\frac{1}{2})+\varepsilon},\forall\varepsilon>0.\]

In 2023, [3] made an improvement for the square-free case and obtained, also conditionally to the Riemann hypothesis, \[\sum_{n\leq x}\mu^{(2)}(n)g_\chi(n)\ll x^{\frac{1}{3}+\varepsilon},\forall\varepsilon>0.\]

More recently, the author [4] proved that \[\sum_{n\leq x}\mu^{(k)}(n)g_\chi(n)\ll x^{\frac{1}{k+1}+\varepsilon},\forall\varepsilon>0,\] conditionally to the generalized Riemann hypothesis.

For a more thoroughly discussion on this and some related problems, as well as the motivation behind the study of modified Dirichlet characters, we refer to [4].

The main purpose of this paper is to analyze limiting distributions of Dirichlet characters and modified Dirichlet characters, over the \(k\)-free integers.

In the above sense, our first result states:

Under these stronger hypothesis, we also obtain a result of similar magnitude than our expected sharp bound (see 1 above).

An active line of research is to study the behaviour of the partial sums of multiplicative functions. A commonality present in some functions that we find in the literature is their connection to the Riemann zeta function.

In most cases, this connection can be readily seen by means of the Euler product formula. A simple calculation shows that the Möbius function can be related to the Riemann zeta function by \[\begin{align} \frac{1}{\zeta(s)}=\sum_{n=1}^\infty\frac{\mu(n)}{n^s}, \end{align}\] for \(\Re(s)>1\). Another example is the generating series of the function \(\mu^{(k)}\): \[\begin{align} \frac{\zeta(s)}{\zeta(ks)}=\sum_{n=1}^\infty\frac{\mu^{(k)}(n)}{n^s}, \end{align}\] for \(\Re(s)>1\). Therefore, the partial sums of these functions are also linked to the Riemann zeta function by employing the Mellin transform (simply making a partial summation) or Perron’s formula.

The functions that we are interested also have a close relation to the Riemann zeta function and to the Dirichlet \(L\)-functions, as we are going to see more carefully later.

Probability viewpoints were implemented by many authors in the study of problems in Analytic Number Theory and arithmetic functions, given its usefulness in providing heuristics arguments and establish precise conjectures. Particularly, limiting distribution has been notorious in the investigation of the possible true order of magnitude of some summatory functions.

We emphasize that conjectures for moments of the Riemann zeta function are essential in our proofs. Specifically, as in [first-hypothesis,second-hypothesis] of 3, we need estimates for the discrete moments: \[J_{r}(T)\mathrel{\vcenter{:}}=\sum_{0<\gamma \leq T}|\zeta'(\rho)|^{2r},\] where \(\rho=\beta+i\gamma\) are the non-trivial zeros of the Riemann zeta function and \(r\in\mathbb{R}\).

In 1989, Gonek (for \(r\leq 0\)) and Hejhal (for \(r>0\)) independently conjectured the following bound:

Later, a restriction on \(r\) arose in the work of Hughes, Keating and O’Connell [9]. In fact, we have a very precise conjecture obtained through a model for the Riemann zeta function using the characteristic polynomial of a random unitary matrix.

Results related to these moments are very limited. Some examples we have are [7], [10]–[12] and [13], for lower and upper bounds, respectively. We also refer to the Introduction of [14], where Humphries gave a nice survey on the subject.

It seems reasonable that a similar conjecture should hold for moments of Dirichlet \(L\)-functions.⁴

In the majority of this paper we assume a weaker hypothesis, namely [first-hypothesis,second-hypothesis]. However, in 6, we may assume 5 and \[\begin{align} \label{conjecture-L} \tilde{J}_{r}(T)\asymp T(\log T)^{(r+1)^2}, \end{align}\tag{1}\] with \(\kappa=1\).

Moreover, our constants will be implicit and therefore we refrain from applying [keating-hughes-oconell-conject,Lfuncwithcte-conj].

As we are going to adress the topic of distribution functions, we briefly comment on this subject. A distribution function \(F:\mathbb{R}\to[0,1]\) satisfies the following properties:

• \(F\) is non-decreasing

• \(\lim_{x\to-\infty}F(x)=0\) and \(\lim_{x\to\infty}F(x)=1\)

• \(F\) is “càdlàg”, i.e., right-continuous and has a limit on the left of each \(x\in\mathbb{R}\).

We also know that, given a distribution function \(F\), there exists a probability \(\mathbb{P}\) related to \(F\), i.e., \(F_{\mathbb{P}}(x)\mathrel{\vcenter{:}}= \mathbb{P}((-\infty,x])\).

Therefore, by showing that \(e^{-\frac{y}{2k}}\sum_{n\leq e^y}f(n)\) has a limiting distribution, one could attempt to extract useful heuristics and informations about the statistical behavior of the partial sums \(\sum_{n\leq e^y}f(n)\).

This was the framework of [16], followed by [14], [15] and then by [17]. We will discuss some tools and steps in 2.

In 6 we also state large deviations results and conjectures, similar to those obtained by [17] in his work on the distribution of the \(k\)-free numbers. These are consequences of the existence of a limiting distribution and the approach taken was primarily studied by [16], for the partial sums of the Möbius function.

1.1 Notation↩︎

If \(f(x)=O(g(x))\) (likewise \(f(x)\ll g(x)\)) we mean that exists a constant \(C>0\) such that \(|f(x)|\leq C g(x)\) for all sufficiently large \(x\). If \(f\asymp g\), then \(g\ll f \ll g\). We write \(f\sim g\) to say that \(f\) is asymptotically equivalent to \(g\), that is, \(\lim_{x\to\infty}\frac{f(x)}{g(x)}=1\). Finally, if \(f(x)=o(g(x))\), then \(|f(x)|\leq \varepsilon g(x)\), for any \(\varepsilon>0\) and all sufficiently large \(x\). In particular, \(f(x)=o(1)\) if \(f(x)\) tends to zero as \(x\to\infty\).

If any of the above asymptotic notations has one or more subindexes, it means that the implicit constant depends on those indicated parameters (e.g. \(O_\varepsilon, \ll_{\varepsilon,k}\)).

Unless stated otherwise, the letter \(p\) will always be a prime number and the product written as \(\prod_p\) means that it runs over all prime numbers.

Finally, for any real number \(x\), the integer part of \(x\) is denote by \([x]\).

2 Discussion of results concerning limiting distributions↩︎

In 2004, [16] proved, under the Riemann Hypothesis and the conjecture \(J_{-1}(T)\ll T\), that \(x^{-\frac{1}{2}}\sum_{n\leq x}\mu(n)\) possess a limiting distribution. With the methods of [18], Ng also studied the tail of this distribution to show large deviation results and conjectures. As a consequence, the following was formulated:

The above is widely believed to be true and the conjectural constant was greatly improved recently [19].

Later, in 2013, [14] proved that the sum of a weighted Liouville function⁵, \[\begin{align} \sum_{n\leq x}\frac{\lambda(n)}{n^\alpha}, 0\leq \alpha <\frac{1}{2}, \end{align}\] has a limiting distribution, also under the Riemann Hypothesis and \(J_{-1}(T)\ll T\).

A year after, [15] proved that, if \(\varphi(y)\) is a \(B^p\)-almost periodic function, that is, for any \(\varepsilon>0\) there exists a real-valued trigonometric polynomial \[\begin{align} P_{N_\varepsilon}(y)=\sum_{n=1}^{N_\varepsilon}r_{n,\varepsilon}e^{i\lambda_{n,\varepsilon}y}, \end{align}\] such that \[\begin{align} \limsup_{Y\to\infty}\frac{1}{Y}\int_0^Y|\varphi(y)-P_{N_\varepsilon}(y)|^p\,dy<\varepsilon^p, \end{align}\] then \(\varphi(y)\) possess a limiting distribution.

As pointed out by the authors, although unpublished, this result was in fact known since 1930s.

Moreover, they used the above to prove a more general statement that, under additional hypothesis, guarantee the existence of limiting distributions for a wide class of summatory functions. Various distributions already known in the literature were obtained as a corollary of their main result. It was also established distributions for error terms of some summatory functions, for example, the sum of weighted Möbius and Liouville function, and the error term for the sum of Möbius function in arithmetic progressions.

In 2017, [17] adapted ideas from [16] and used results of [15] to prove that the error term of \(\sum_{n\leq x}\mu^{(k)}(n)\), after a normalization, also has a limiting distribution. More precisely, under the Riemann Hypothesis and \(J_{-1}(T)\ll T^{1+\varepsilon}\), \[\begin{align} x^{-\frac{1}{2k}}M_k(x)=x^{-\frac{1}{2k}}\bigg(\sum_{n\leq x}\mu^{(k)}(n)-\frac{x}{\zeta(k)}\bigg), \end{align}\] possess a logarithmic limiting distribution.

More recently, [19] employed some tools used by [17] and improved the main result of [15] (namely, conditions for a function to be \(B^2\)-almost periodic - see 12 in the next section), weakening some conditions and stating its theorems in a more general form.

We refer to [15] and the references therein for a more complete historical overview on the subject of limiting distributions.

3 Preliminaries↩︎

Let \(F(s)\) be the Dirichlet series with coefficients \(f=\mu^{(k)}\chi\), where \(\chi\) is a non-principal Dirichlet character modulo \(q\) and \(k\geq 2\) a fixed integer. By the Euler product formula, \[\begin{align} F(s)=\frac{L(s,\chi)}{L(ks,\chi^k)}, \end{align}\] for \(\Re(s)>1\).

We focus mostly on quadratic characters and in this case, for \(\Re(s)>1\), we have \[\begin{align} F(s)= \begin{dcases} \frac{L(s,\chi)P(ks)}{\zeta(ks)} &\text{ if k is even,}\\ \frac{L(s,\chi)}{L(ks,\chi)} &\text{ if k is odd,} \end{dcases} \end{align}\] where \(P(s)\mathrel{\vcenter{:}}=\prod_{p\mid q}(1-\frac{1}{p^s})^{-1}\).

Moreover, when working with modified Dirichlet characters we are implicitly assuming that \(\chi\) is real. If we let \(G(s)\) be the generating Dirichlet series of \(\mu^{(k)}g_\chi\), then we can write \[\begin{align} G(s)= \begin{dcases} \frac{L(s,\chi)P(s)}{\zeta(ks)} &\text{ if k is even,}\\ \frac{L(s,\chi)}{L(ks,\chi)}\frac{P(s)}{P(ks)} &\text{ if k is odd,} \end{dcases} \end{align}\] where \(P(s)\) is the same as defined above and \(\Re(s)>1\).

Given this representation, it becomes clear that our functions relates to \(\zeta(s)\) and \(L(s,\chi)\). We thus discuss two classical results contained in the literature: the first one is regarding the functional equation for these functions and the second one being the analogue, for Dirichlet \(L\)-functions, of the classical von Mangoldt estimate for the number of non-trivial zeros of the Riemann zeta function.

Recalling the asymmetric functional equation for the Riemann zeta function, expressed with the ratio of Gamma function factor, \[\begin{align} \zeta(s)=\pi^{s-\frac{1}{2}}\frac{\Gamma(\frac{1-s}{2})}{\Gamma(\frac{s}{2})}\zeta(1-s), \end{align}\] we also have the following for Dirichlet \(L\)-functions asssociated to a Dirichlet character \(\chi\) of modulus \(q\): \[\begin{align} \label{functional-eq-L} L(s,\chi)=\frac{\tau(\chi)}{i^\delta \sqrt{q}}\Delta(s)L(1-s,\overline{\chi}), \end{align}\tag{2}\] where \(\tau(\chi)=\sum_{n=1}^q\chi(n)e^{2\pi in/q}\), \[\begin{align} \delta\mathrel{\vcenter{:}}= \delta(\chi)= \begin{cases} 0 &\text{ if \chi(-1)=1,}\\ 1 &\text{ if \chi(-1)=-1}, \end{cases} \end{align}\] and \(\Delta(s)\) is the ratio of gamma factors⁶, that is, \[\begin{align} \Delta(s)= \begin{cases} \pi^{s-\frac{1}{2}}\Gamma(1-\frac{s}{2})/\Gamma(\frac{s+1}{2}) &\text{ if \delta=1,}\\ \pi^{s-\frac{1}{2}}\Gamma(\frac{1-s}{2})/\Gamma(\frac{s}{2}) &\text{ if \delta=0.} \end{cases} \end{align}\]

A bound that we are going to use recurrently follows from an application of Stirling’s formula. One can show that \[\begin{align} \label{gamma-factor-bound} |\Delta(\sigma+it)|\asymp |t|^{\frac{1}{2}-\sigma}. \end{align}\tag{3}\]

From this, we have

Clearly a similar result holds for \(L(s,\chi)\).

Now onto the second observation. As usual, let \(N(T)\) be the number of non-trivial zeros of the Riemann zeta function in the strip \((0,1)\) and up to imaginary height \(T\). Von Mangoldt proved that \[\begin{align} N(T)=\frac{T}{2\pi}\log\frac{T}{2\pi e}+O(\log T). \end{align}\]

Similar to this, we also have:

In particular, both \(N(T)\) and \(N(T,\chi)\) are \(\sim C_q T\log T\).

3.1 Ng’s result and the existence of a limiting distribution↩︎

Now we formalize the ideas mentioned in 2 concerning the existence of a limiting distribution for a class of summatory functions and present a proposition that is central in our proof.

Before continuing to these result, we first fix a few definitions and consider some hypothesis, similar to Ng’s paper [19].

We have the following general sequences:

(i) \(\boldsymbol{\lambda}=(\lambda_n)_{n\in\mathbb{N}}\) is a non-decreasing sequence of positive numbers.

(ii) \(\boldsymbol{r}=(r_n)_{n\in\mathbb{N}}\) is a sequence of complex numbers.

We also define \(N_{\boldsymbol{\lambda}}(T)\) as the number of \(n\in\mathbb{N}\) such that \(\lambda_n\leq T\).

We are going to need the following proposition which is a generalization of Lemma 5 of [17]:

The succeding is essential in our main theorem.

3.2 Further discussion about \(P(s)\)↩︎

Here we are going to examine more thoroughly the finite product \[P(s)\mathrel{\vcenter{:}}=\prod_{p\mid q}\bigg(1-\frac{1}{p^s}\bigg)^{-1},\] defined above, that depends on the fixed integer \(q\geq 3\).

Note that, for each \(p\mid q\), this function has simple poles at the pure imaginary numbers: \[\begin{align} \label{zeros-P} z_p(t)=i\frac{2\pi t}{\log p}, t\in\mathbb{Z}\backslash\{0\}, \end{align}\tag{4}\] and a pole of order \(\omega(q)\) at \(s=0\), where \(\omega(q)\) is the number of distinct prime factors of \(q\).

However, in the half-plane \(\{s\in\mathbb{C} : \Re(s)>0\}\), we see that \(P(s)\) is analytic. Moreover, if we fix any \(\sigma_0>0\), then \[\begin{align} \label{bound-for-P40s41} P(s)=\prod_{p\mid q}\bigg(1-\frac{1}{p^s}\bigg)^{-1}\ll \prod_{p\mid q}\bigg(1-\frac{1}{p^{\sigma_0}}\bigg)^{-1}\ll\prod_{p\mid q}p^{\sigma_0}\ll q^{\sigma_0}\ll 1, \end{align}\tag{5}\] for all \(\Re(s)\geq \sigma_0>0\). This fact will be used in the proofs of upcoming lemmas.

Further, as we saw, the generating series of \(\mu^{(k)}\chi\) and \(\mu^{(k)}g_\chi\) are very similar: for even \(k\), the former has the factor \(P(ks)\), while the latter has \(P(s)\). Thus, by the above considerations, we remark that our proofs should hold similarly for both functions.

If \(k\) is an odd positive integer, recall that we have the extra factor \(P(s)/P(ks)\) in the series involving the modified characters, whereas its counterpart is represented only by a product of \(L\)-functions. As we are going to see, this difference between them should not impose any extra difficulties.

The basic reason for this last assertion is that we aim to avoid the poles of \(P(s)\) in our computations, since contributions coming from the residues at those points are too large when the modulus \(q\) is not a prime power. Therefore, we are mostly going to work in the half-plane \(\Re(s)>0\).

4 Main lemmas↩︎

The first lemma we are going to prove is for any primitive Dirichlet character. We have the following:

We prove the preceding lemma using 11.

An important tool for us is Perron’s formula and the version we are going to apply is from [20].

Before we proceed further, the next technical lemma will be needed.

Combining the two previous results with the Residue Theorem and Cauchy’s Integral Formula Theorem, we show an explicit representation for the partial sums of our functions of interest.

For a general imaginary height \(T\), we assume more than the simplicity of zeros and also must pay the price of an additional error term. We have

5 Proof of [log-meas-zero-thm,cor-distribution]↩︎

Now we use Ng’s result (12) to prove our theorem on the logarithmic distribution of \(x^{-\frac{1}{2k}}\sum_{n\leq x}f(n)\).

Next we prove 4. As we remarked earlier, this result essentially shows, under stronger conditions, that the bound \(\ll x^{\frac{1}{2k}+\varepsilon}\) for the partial sums of \(f=\mu^{(k)}\chi\) (resp. \(\mu^{(k)}g_\chi\)) is attainable.

6 Large deviations↩︎

In this section we assume⁷ \(f=\mu^{(k)}\chi\) and the following conjectures:

That is to say, if we take \(\gamma_1,\dots,\gamma_n\) positive imaginary parts of non-trivial zeros of the Riemann zeta function (resp. Dirichlet \(L\)-function), we have that \[\begin{align} a_1\gamma_1+\cdots+a_n\gamma_n =0 \end{align}\] only if \(a_i=0,\forall i\).

Let \(V\in\mathbb{R}\) be fixed and define \[\begin{align} g(x)= \begin{cases} 1 &\text{ if x\geq V,}\\ 0 &\text{ otherwise.} \end{cases} \end{align}\]

From 2, in the case where \(\nu_k\) is absolutely continuous, our 3 implies \[\begin{align} \lim_{Y\to \infty}\frac{1}{Y}\mathop{\mathrm{meas}}\bigg\{y\in[0,Y] : \frac{\sum_{n\leq e^y}f(n)}{e^{\frac{y}{2k}}}\geq V\bigg\}=\int_{V}^\infty d\nu_k(x)=\nu_k([V,\infty)), \end{align}\] where \(\mathop{\mathrm{meas}}\{\cdot\}\) is the Lebesgue measure on \(\mathbb{R}\). Roughly, we would like to study how often \(\sum_{n\leq e^{y}}f(n)\) takes values greater than \(e^{\frac{y}{2k}}V\).

Therefore, defining a proper random variable \(X(\boldsymbol{\theta})\) (that comes from the construction of the limiting distribution \(\nu_k\) above), we would like to estimate bounds for \[\begin{align} \nu_k([V,\infty))=\mathbb{P}(X(\boldsymbol{\theta})\geq V). \end{align}\]

Turns out this is identical to the upper and lower bounds obtained by Meng for the distribution of \(k\)-free numbers, so we just highlight here the results and main concepts of the proof.

The following proposition shows that the Fourier transform of \(\nu_k\) can be explicitly represented, as long as we assume the LI/GLI conjecture.

The proof follows similarly to Corollary 1 of [16] or Theorem 6.1 of [14] and the tool that allows us to compute this Fourier transform is the construction of \(\nu_k\) by means of the Kronecker-Weyl Theorem [14].

Now, as in the work of [16] and [17], we let \(\boldsymbol{\theta}=(\theta_1,\theta_2,\dots)\in\mathbb{T}^\infty\). Assuming the LI/GLI conjectures, the limiting distribution \(\nu_k\) is equal to \(\nu_{k,X}(x)=\mathbb{P}(X^{-1}(-\infty,x))\), where \(X\) is the random variable on the infinite torus defined by \[\begin{align} \label{rv-even} X(\boldsymbol{\theta})= \begin{dcases} 2\sum_{\gamma>0}\bigg|\frac{L(\frac{\rho}{k},\chi)P(\rho)}{\rho\zeta'(\rho)}\bigg|\sin(2\pi \theta_\gamma) &\text{ if k is even,}\\ 2\sum_{\gamma>0}\bigg|\frac{L(\frac{\rho}{k},\chi)}{\rho L'(\rho,\chi)}\bigg|\sin(2\pi\theta_\gamma) &\text{ if k is odd.} \end{dcases} \end{align}\tag{8}\]

Thus, under these assumptions, one can study \(\nu_k\) by means of the above random variable \(X(\boldsymbol{\theta})\) defined on the infinite torus \(\mathbb{T}^\infty\).

The main tool employed by previous authors to work with sine random variables as in 8 is the following result of [18]:

By letting \(r_\gamma=2|\frac{L(\rho/k,\chi)P(\rho)}{\rho\zeta'(\rho)}|\) (resp. \(r_\gamma=2|\frac{L(\rho/k,\chi)}{\rho L'(\rho,\chi)}|\)), similar to Sections 4 and 5 of [17], we obtain

The more cautious analysis that leads to ?? gives support to a conjecture that equates the Large Deviation Conjecture in [17], i.e., the right hand side gives the precise order of \(\nu_k([V,\infty))\). Additionally, as a consequence of this and the analysis of Ng [16], we have the following:

6.1 Main ideas of the proof of 19↩︎

To prove the first statement ?? , considering 8 , we take \(r_\gamma=2|\frac{L(\rho/k,\chi)P(\rho)}{\rho\zeta'(\rho)}|\) and our aim is to apply Montgomery’s Lemma.

From 3 and from 1 combined with Cauchy’s Integral Formula, we have that \[\bigg|L\bigg(\frac{\rho}{k},\chi\bigg)\bigg|\gg |\gamma|^{\frac{1}{2}-\frac{1}{2k}-\varepsilon}\quad\text{ and }\quad\frac{1}{|\zeta'(\rho)|}\gg \frac{1}{|\gamma|^\varepsilon},\] respectively.

The hypothesis of the theorem and the above estimates gives the upper and lower bounds: \[T^{\frac{1}{2}-\frac{1}{2k}-\varepsilon}\ll \sum_{0<\gamma\leq T}r_\gamma\ll T^{\frac{1}{2}-\frac{1}{2k}+\varepsilon}\quad\text{ and }\quad \frac{1}{T^{\frac{1}{k}+\varepsilon}}\ll\sum_{\gamma>T}r_\gamma^2\ll \frac{1}{T^{\frac{1}{k}-\varepsilon}}.\]

Now we apply 6 to obtain \[\exp(-c_1T^{1+\varepsilon})\leq \nu_k\bigg(\bigg[2\sum_{0<\gamma \leq T}r_\gamma,\infty\bigg)\bigg)\leq \exp(-c_2T^{1-\varepsilon}),\] for constants \(c_1,c_2>0\) and conclude by taking \(V=T^{\frac{1}{2}-\frac{1}{2k}}\).

To prove the second part of 19, namely ?? , we need the following result from [17]: under the Riemann hypothesis, for \(w\in(0,1)\) and \(r\geq 1\), we have \[\sum_{0<\gamma\leq T}|\zeta(1-w\rho)|^{2r}\ll T\log T,\] and \[\sum_{0<\gamma\leq T}\frac{1}{|\zeta(1-w\rho)|^{2r}}\ll T\log T.\]

With the above at our disposal, we use Hölder’s inequality and the conjectural bounds for the \(2l\)-th moments to estimate the sums over \(r_\gamma\) and \(r_\gamma^2\). By following Meng’s steps, we are able to produce the extra logarithmic term in the denominator of ?? .

7 Additional results↩︎

This section is dedicated to present theorems that are somewhat similar to the Weak Mertens Hypothesis, which asserts that, \[\begin{align} \label{weak-mertens-conj} \int_1^X\bigg(\frac{\sum_{n\leq x}\mu(n)}{x}\bigg)^2\,dx\ll \log X. \end{align}\tag{9}\]

The above is a notably strong conjecture and, in particular, 9 implies the Riemann hypothesis, that all the zeros of \(\zeta\) are simple, the bound \(\zeta'(\rho)^{-1}\ll |\rho|\) and the convergence of the sum \(\sum_\gamma |\rho\zeta'(\rho)|^{-2}\) [22].

Our result states:

As a consequence of the existence of a limiting distribution, by Theorem 1.14 of [15] (also see [16] and [17]), we obtain a stronger result, namely: \[\begin{align} \int_{2}^X\bigg(\frac{\sum_{n\leq x}f(n)}{x^{\frac{1}{2k}}}\bigg)^2\,\frac{dx}{x}\sim \beta_k \log X, \end{align}\] where, for \(f=\mu^{(k)}\chi\), \[\begin{align} \beta_k= \begin{dcases} 2\sum_{\gamma>0}\bigg|\frac{L(\frac{\rho}{k},\chi)P(\rho)}{\rho\zeta'(\rho)}\bigg|^2 &\text{ if k is even,}\\ 2\sum_{\gamma>0}\bigg|\frac{L(\frac{\rho}{k},\chi)}{\rho L'(\rho,\chi)}\bigg|^2 &\text{ if k is odd.} \end{dcases} \end{align}\] As usual, if \(f=\mu^{(k)}g_\chi\), we replace the factor \(P(\rho)\) by \(P(\rho/k)\) if \(k\) is even and include \(P(\rho/k)/P(\rho)\) if \(k\) is odd.

At last, note that \(\sum_{0<\gamma\leq T}|\zeta'(\rho)|^{-2}\ll T^{1+\frac{1}{k}-\varepsilon}\) (resp. \(\sum_{0<\gamma\leq T}|L'(\rho,\chi)|^{-2}\ll T^{1+\frac{1}{k}-\varepsilon}\)) can be used to show the convergence of \(\beta_k\) above.

8 Final remarks↩︎

Throughout this manuscript we assumed that the second negative moment for \(\zeta(s)\) and \(L(s,\chi)\) has cancellation \(\ll T^{1+\frac{1}{k}-\varepsilon}\) and, for 4, we obtained a cancellation of \(\ll T^{1+\frac{1}{2k}-\frac{\varepsilon}{2}}(\log T)^{\frac{1}{2}}\) for the first negative moment, as a consequence of the former. However, for some summatory functions, e.g. partial sums of the Möbius function, as pointed out by [17] (also covered in the recent work of [19]), one can assume the much weaker bound \(\sum_{0<\gamma\leq T}|\zeta'(\rho)|^{-2}\ll T^{2-\varepsilon}\).

Assuming the Riemann hypothesis and the simplicity of zeros, we note that we have \(\sum_{0<\gamma\leq T}|\zeta'(\rho)|^{-2}\ll T^{2+\varepsilon}\), since \(|\zeta'(\rho)|^{-1}\ll |\rho|\). Furthermore, the Weak Mertens Hypothesis 9 implies that \[\sum_{0<\gamma\leq T}\frac{1}{|\zeta'(\rho)|^{2r}}= \begin{cases} o(T^{r+1}(\log T)^{1-r}) &\text{ if 0<r<1,}\\ o(T^{2r}) &\text{ if r\geq 1.} \end{cases}\] This result follows from Hölder’s inequality and the fact that \(\sum_{\gamma}|\rho\zeta'(\rho)|^{-2}\) is convergent under this conjecture. We refer to [23] for a proof of the above and, although the authors considered the sum where \(\gamma\in (T,2T]\), note that the same proof is carried for sums over all zeros of \(\zeta\) with ordinates in \((0,T]\).

Also in [23], assuming only the Riemann hypothesis, [23] showed that \[\begin{align} \label{Bui-Florea-Milinovich-result} \sum_{\gamma\in \mathcal{F}}\frac{1}{|\zeta'(\rho)|^{2r}}\ll_{r,\delta} \begin{cases} T^{1+\delta} &\text{ if r<\frac{1}{2},}\\ T^{r+\frac{1}{2}+\delta} &\text{ if r\geq \frac{1}{2},} \end{cases} \end{align}\tag{10}\] for any \(\delta>0\), where \[\mathcal{F}=\bigg\{\gamma\in (T,2T] : |\gamma-\gamma'|\gg \frac{1}{\log T} \text{ and \gamma' is any other ordinate of a zero of \zeta}\bigg\}.\]

As remarked by the authors, the set \(\mathcal{F}\) is expected to be arbitrarily close to full density inside the dyadic interval \((T,2T]\). Thus, if such result could be extended to sums over all ordinates \(\gamma\in (0,T]\), only the generalized Riemann hypothesis would be enough to prove our results, since [first-hypothesis,second-hypothesis] from 3 would become obsolete.

In prior works related to limiting distribution of summatory functions, the strategy for establishing explicit formulas, as in 4, was to use Cauchy’s Integral Formula and the Residue Theorem in a rectangle of vertices \(c\pm iT\) and \(-U\pm iT\), for some \(c,U>0\), estimate the integrals over each line segment and take \(U\to \infty\). Our case is more delicate due to the function \(P(s)\). If we go beyond the imaginary axis, the contribution coming from the poles described in 4 would be too large, as discussed in 3.2. This explains why we maintained our contour in the half-plane \(\Re(s)>0\).

The usual approach described above works when we consider the case where \(k\) is odd and we are working with the function \(\mu^{(k)}\chi\), since the representation of its Dirichlet series does not contain the factor \(P(s)\), as presented in 3.

9 Acknowledgements↩︎

I would like to thank Marco Aymone for first pointing out the problem, his support and for reading an early version. I am also thankful for Professor Winston Heap for his interest on the topic and also suggesting the problem, and helpful discussions while I was visiting the Max Planck Institute for Mathematics. Moreover, I thank the MPIM for the warm hospitality. Finally, I would like to thank Professor Micah Milinovich for providing a reference that contributed to the improvement of the exposition.

References↩︎

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1. This project is supported in part by the Coordenação de Aperfeiçoamento de Pessoal de Nível Superior - Brasil (CAPES), FAPEMIG Grant Universal No. APQ-00256-23 and CNPq Grant Universal No. 403037/2021-2.↩︎

2. In the sense that \(f\) is supported on the square-free integers and takes values \(\pm 1\) at prime numbers.↩︎

3. We say that a set \(\mathcal{A}\) has finite logarithmic measure if \(\int_{\mathcal{A}\cap [1,\infty)}\,\frac{dx}{x}<\infty\).↩︎

4. See also [15], where the authors assume a similar type of conjectural bound.↩︎

5. The Liouville function \(\lambda(n)\) is defined as the completely multiplicative function that takes value \(-1\) at each prime number.↩︎

6. Usually the ratio is represented by the letter \(\chi\), but we use \(\Delta\) to avoid confusion with the Dirichlet characters.↩︎

7. The same justifications from previous sections can be made to convince the reader that the results in this section also holds for \(f=\mu^{(k)}g_\chi\). To avoid excessive repetition, here we won’t expand for the modified character.↩︎